Median, Mode, and Range

5.5 Median, Mode, and Range

Describe situations in real life where the mean STATES is not a good representation of the average. STANDARDS MA.6.S.6.1 1 ACTIVITY: Comparing Three Samples

Work with a partner. Surveys are taken in three grade 6 –12 schools in Florida. Make up a story about the three surveys. Find the mean for each survey. Do you think the mean is a good way to represent the “average” of each group? Why or why not? a.

1112 13 14 15 16 17 18

216 Chapter 5 Ratios, Rates, and Data Analysis 2 ACTIVITY: When the Mean is Misleading

Work with a partner. Read and re-read each statement. Think of a better way to represent the “average” so that the statement is not so misleading. a. Someone is trying to criticize a small high school by saying Age of Graduating Class “Last year, the average age of the 12 graduating class was 22 years old.” 10 When you look into the facts, you 8 ﬁ nd that the class had a senior citizen who went back to school to 6 earn a diploma. Here are the ages 4

for the class. 2

18, 18, 18, 18, 18, 17, 18, 0

Number of Graduates 74 19, 18, 18, 18, 18, 18, 74 17 19 18 Age of Graduates What percent of the ages are below the mean? b. There is a small town where Families’ Yearly Income most of the people are having a difﬁ cult time getting by because 5 of low incomes. Someone is trying to ignore the problem and writes 4 an article in the newspaper

saying “It is not so bad in the 3 town. The average income for a family is $52,000 a year.” Here 2 are the incomes.

$20,000, $20,000, $20,000, Number of Families $20,000, $30,000, $30,000, 1 $30,000, $30,000, $40,000, 0 $40,000, $40,000, $50,000, 50,000 310,000 20,000 $50,000, $50,000, $310,000 30,000 40,000 What percent of the families have Yearly Income (dollars) incomes below the mean?

3. IN YOUR OWN WORDS Describe situations in real life where the mean is not a good representation of the average. What measures (other than mean) can you use to describe an average?

Use what you learned about the mean to complete Exercises 5 and 6 on page 220.

Section 5.5 Median, Mode, and Range 217 5.5 Lesson Lesson Tutorials

A measure of central tendency is a measure that represents the center of Key Vocabulary a data set. The mean is one type of measure of central tendency. Here are measure of central two others. tendency, p. 218 median, p. 218 mode, p. 218 range, p. 219 Median Words Order the data. For a set with an odd number of values, the median is the middle value. For a set with an even number of values, the median is the mean of the two middle values. Numbers Data: 5, 8, 9, 12, 14 The median is 9.

Data: 2, 3, 5, 7, 10, 11 5 + 7 The median is — , or 6. Study Tip 2 The mode is the only measure of central Mode tendency that can be Words The mode of a data set is the value or values that occur most used to describe a set often. Data can have one mode, more than one mode, or no of data that is not made mode. When all values occur only once, there is no mode. up of numbers. Numbers Data: 11, 13, 15, 15, 18, 21, 24, 24

The modes are 15 and 24.

EXAMPLE 1 Finding the Median and Mode

Find the median and mode of the bowling scores. Bowling Scores 120 135 160 125 90 90, 105, 120, 125, 135, 145, 160, 160, 175, 205 Order the data. 205 160 175 105 145 135 + 145 280 Median: — = — , or 140 Add the two middle values and divide by 2. 2 2

Mode: 90, 105, 120, 125, 135, 145, 160, 160, 175, 205

The value 160 occurs most often.

The median is 140. The mode is 160.

Find the median and mode of the data. 1. 20, 4, 17, 8, 12, 9, 5, 20, 13 2. 100, 75, 90, 80, 110, 102

218 Chapter 5 Ratios, Rates, and Data Analysis EXAMPLE 2 Finding the Mode

The list shows the favorite types of movie for students in a class. Favorite Types of Movies Organize the data in a frequency table. Then ﬁ nd the mode. Comedy Drama Horror Type Tally Frequency Horror Drama Horror ∣̇∣̇∣̇∣̇∣ ̇ The number of tally Comedy Comedy Action Action ̇̇̇̇ 5 ∣̇∣̇∣̇∣̇∣ ̇∣̇∣̇∣ marks is the frequency. Action Comedy Action Comedy ̇̇̇̇ ̇̇̇ 8 ̇∣̇∣̇∣̇∣ Horror Drama Comedy Drama ̇̇̇̇ 4 ∣̇∣̇∣̇∣̇∣ ̇∣̇∣ Comedy Comedy Horror Horror ̇̇̇̇ ̇̇ 7 Horror Comedy Action Make a tally for each vote. Horror Action Drama Comedy received the most votes. So, the mode is comedy.

3. In Example 2, one member of the class was absent and Exercises 7–15 ends up voting for horror. Does this change the mode? Explain.

The range of a data set is the difference between the greatest value and the least value. The range describes how spread out the data are.

EXAMPLE 3 Standardized Test Practice

Record Length The table shows the record lengths of the six venomous Snake (inches) snakes found in Florida. What is the range of the lengths? Copperhead 53 ○A 43.5 inches ○B 48.5 inches Cottonmouth 74.5 ○C 65 inches ○D 74.5 inches Diamondback 96 rattlesnake 31, 47.5, 53, 74.5, 74.5, 96 Order the dataata ffromrom least to ggreatest.r Timber rattlesnake 74.5 The least value is 31. The greatestt Pygmy rattlesnake 31 value is 96. So, the range is 96 − 31,1, Coral snake 47.5 or 65 inches.

The correct answer is ○C .

4. The data set shows how many inches of rain fell in several cities during a storm. 3.9, 4.25, 4.1, 3.7, 3.8, 4.3, 3.8, 4.5 Find the range of the data.

Section 5.5 Median, Mode, and Range 219 5.5 Exercises Help with Homework

1. NUMBER SENSE Give an example of a data set that has no mode. 2. WRITING Which is affected more by an outlier: the mean or the median? Explain. 3. REASONING What two data values do you need to know to ﬁ nd the range? Explain. 4. NUMBER SENSE A data set has a mean of 7, a median of 5, and a mode of 8. Which of the numbers 7, 5, and 8 must be in the data set? Explain.

6)=3 9+(- )= +(-3 3 9)= 4+(- 1)= 9+(-

Find the mean of the data. Is the mean a good “average”? If not, what would be a better “average”? Explain. 5. 8, 29, 30, 30, 30 6. 16, 24, 13, 66, 22, 26, 22, 28

Find the median, mode(s), and range of the data. 1 3 7. 3, 5, 7, 9, 11, 3, 8 8. 14, 19, 16, 13, 16, 14 9. 93, 81, 94, 71, 89, 92, 94, 99 10. 44, 13, 36, 52, 19, 27, 33 11. 12, 33, 18, 28, 29, 12, 17, 4, 2 12. 55, 44, 40, 55, 48, 44, 58, 67

13. ERROR ANALYSIS Describe and correct the error in ﬁ nding the median and range of the data. ✗ The median is 58.

63, 55, 49, 58, 50, 59, 51

The range is 63 − 51, or 12.

Find the mode(s) of the data.

2 14. Shirt Color 15. Talent Show Acts Black Blue Red Singing Dancing Comedy Pink Black Black Singing Singing Dancing Gray Green Blue Juggling Dancing Singing Blue Blue Red Singing Poetry Dancing Yellow Blue Blue Comedy Magic Dancing Black Orange Black Poetry Singing Singing Black

16. REASONING In Exercises 14 and 15, can you ﬁ nd the mean and median of the data? Explain. 17. REASONING How does an outlier affect the range of a data set? Explain.

220 Chapter 5 Ratios, Rates, and Data Analysis 18. SUPREME COURT The data show the years of service of the justices of the Supreme Court as of 2008. Find the mean, median, mode(s), and range of the data. 2, 14, 15, 20, 3, 22, 18, 33, 17

Find the mean, median, mode(s), and range of the data. 19. 5, 13, 24, 28, 5, 19, 20, 16, 0, 19 20. 77, 73, 75, 78, 79, 72, 74, 68, 81, 83, 76

1 5 1 3 5 1 5 1 21. 4.7, 8.51, 6.5, 7.42, 9.64, 7.2, 9.3 22. 8 — , 6 — , 3 — , 5 — , 6 — , 5 — , 10 — , 4 — 2 8 8 4 8 4 8 2

Little League Team 23. BASEBALL Find the median, mode, and range

8 of the ages of players on the Little League team. 7 Explain how you found your answers. 6 24. HOMEWORK 5 Write down the number of 4 minutes you spent on homework each day 3 of the past week. 2 a. Find the mean, median, mode(s), and

Number of Players 1 0 range of the data. 10 11 12 Age b. How do you think your success as a student is related to the median of your data? the mode? the range?g

25. MOTOCROSS The ages of the racers in a bicycle motocross race aree 14, 22, 20, 25, 26, 17, 21, 30, 27, 25, 14, and 29. The 30-year-old dropsps out of the race and is replaced with a 15-year-old. How are the mean,an, median, mode, and range of the ages affected?

26. REASONING The median of the data is 37 and the range is 29. What is the missing value? Explain how you found your answer.

48, 56, 29, , 42, 27

27. Write a set of data with six values that has a mean of 28, a median of 29, and a range of 18.

Identify the outlier. Find the mean of the data set with and without the outlier. 28. 22, 25, 17, 27, 29, 83, 26, 20, 20, 21 29. 64, 71, 58, 60, 65, 35, 70, 74

30. MULTIPLE CHOICE A parking lot contains 18 vans, 26 SUVs, and 24 cars. What is the ratio of vans to the total number of vehicles? 9 8 9 3 ○A — ○B — ○C — ○D — 25 17 34 4

Section 5.5 Median, Mode, and Range 221